**INSTRUCTIONS:**

- Solve all the questions using MS WORD.
- Make use to use Math Equation Editor where applicable
- Copy your excel solutions and paste in document
- Submit completed assignment through Moodle. No email submissions allowed
- Make sure to submit on time. No late submissions allowed

**INDIVIDUAL PROBLEM#5 56- Marks**

Question 1 – 16- Marks

Consider the following linear program:

Min 8X + 12Y

s.t.

1X + 3Y ≥ 9

2X + 2Y ≥10

6X + 2Y ≥ 18

X, Y 0

- Use the graphical solution procedure to find the optimal solution.
- Assume that the objective function coefficient for X changes from 8 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.
- Assume that the objective function coefficient for X remains 8, but the objective function coefficient for Y changes from 12 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.
- The sensitivity report for the linear program in part (a) provides the following objective coefficient range information:

Variable | Objective Coefficient | Allowable Increase | Allowable Decrease |

X | 8.000 | 4.000 | 4.000 |

Y | 12.000 | 12.000 | 4.000 |

How would this objective coefficient range information help you answer parts (b) and (c) prior to resolving the problem?

Question 2 – 16 Marks

Consider the following linear program:

Min 8X + 12Y

s.t.

1X + 3Y ≥ 9

2X + 2Y ≥10

6X + 2Y ≥ 18

X, Y 0

Suppose that the right-hand side for constraint 1 is increased from 9 to 10.

- Use the graphical solution procedure to find the optimal solution.
- Use the solution to part (a) to determine the shadow price for constraint 1.
- The sensitivity report for the linear program in this question provides the following right- hand-side range information:
- The shadow price for constraint 2 is 3. Using this shadow price and the right-hand- side range information in part (c), what conclusion can be drawn about the effect of changes to the right-hand side of constraint 2?

Constraint | Constraint
RH Side |
Allowable Increase | Allowable Decrease |

1 | 9.000 | 2.000 | 4.000 |

2 | 10.000 | 8.000 | 1.000 |

3 | 18.000 | 4.000 | Infinite |

What does the right-hand-side range information for constraint 1 tell you about the shadow price for constraint 1?

Question 3 – 16 Marks

Chrystab Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A portfolio consists of *U* shares of U.S. Oil and *H* shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The portfolio risk index (0.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 1000 shares of U.S. Oil. The linear programming formulation that will maximize the total annual return of the portfolio is as follows:

Max 8*U* + 5*H*Maximize total annual return

s.t.

25*U* + 50*H* ≤80,000Funds available

0.50*U*+ .25*H* ≤ 700 Risk Maximum

1*U*≤ 1000U.S. Oli Maximum

*U, H*≤ 0

The sensitivity report for this problem is shown in below in Table 1.

Table 1

Variable Cells |
||||||

Model Variable | Name | Final Value | Reduced Cost | Objective Coefficient | Allowable Increase | Allowable Decrease |

U |
U.S Oil | 800.000 | 0.000 | 3.000 | 7.000 | 0.5000 |

H |
Huber | 1200.000 | 0.000 | 5.000 | 1.000 | 3.5000 |

Constraints |
||||||

Constraint Number | Name | Final Value | Shadow Price | Constraint R.H. Side | Allowable Increase | Allowable Decrease |

1 | Funds available | 80000.000 | 0.093 | 80000.000 | 60000.000 | 15000.000 |

2 | Risk maximum | 700.000 | 1.333 | 700.000 | 75.000 | 300.000 |

3 | U.S. Oil maximum | 800.000 | 0.000 | 1000.000 | 1E+30 | 200.000 |

- What is the optimal solution, and what is the value of the total annual return?
- Which constraints are binding? What is your interpretation of these constraints in terms of the problem?
- What are the shadow prices for the constraints? Interpret each.
- Would it be beneficial to increase the maximum amount invested in U.S. Oil? Why or why not?

Question 48 Marks

The New West Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year’s program. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown:

Constraints |
Television |
Radio |
Online |

Audience per advert | 1000,000 | 18000 | 40,000 |

Cost per Advert | $2000 | $300 | $600 |

Maximummedia usage | 10 | 20 | 10 |

To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized.

- If the promotional budget is limited to $18,200, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media, and what is the total audience reached?
- By how much would audience contact increase if an extra $100 were allocated to the promotional budget?

INDIVIDUAL PROBLEM #6 56 Marks |

**Question 1 – 15 – Marks**

Consider the following all-integer linear program:

Max1 x_{1+ } 1x_{2 }

s.t.

4×1 + 16x_{2}22

1×1 + 15×2 15

2x_{1+}1x_{2} 9

x_{1}, x_{2} 0 and integer

- Graph the constraints for this problem. Use dots to indicate all feasible integer solutions.
- Solve the LP Relaxation of this problem.
- Find the optimal integer solution

The optimal solution to the LP Relaxation is shown on the above graph to be *x*1 = 4, *x*2 = 1.Its value is 5.

The optimal integer solution is the same as the optimal solution to the LP Relaxation.This is always the case whenever all the variables take on integer values in the optimal solution to the LP Relaxation.

Question 2 – 16- Marks

Hawkins Manufacturing Company produces connecting rods for 4- and 6-cylinder auto- mobile engines using the same production line. The cost required to set up the production line to produce the 4-cylinder connecting rods is $2000, and the cost required to set up the production line for the 6-cylinder connecting rods is $3500. Manufacturing costs are $15 for each 4-cylinder connecting rod and $18 for each 6-cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If there is a production changeover from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 6000 6-cylinder connecting rods and 8000 4-cylinder connecting rods. Let

x_{4} = the number of 4-cylinder connecting rods produced next week

x_{6}= the number of 6-cylinder connecting rods produced next week.

S_{4} = 1 if the production line is set up to produce the 4-cylinder connecting rods; 0 if

otherwise

S_{6} = 1 if the production line is set up to produce the 6-cylinder connecting rods; 0 if

otherwise

- Using the decision variables x
_{4}and s_{4}, write a constraint that limits next week’s pro- duction of the 4-cylinder connecting rods to either 0 or 8000 units. - Using the decision variables X
_{6}and S_{6}, write a constraint that limits next week’s pro- duction of the 6-cylinder connecting rods to either 0 or 6000 units. - Write three constraints that, taken together, limit the production of connecting rods for next week.
- Write an objective function for minimizing the cost of production for next week.

Question 3 – 15 – Marks

Consider again the Ohio Trust Inc. problem described in Problem 15. Suppose only a limited number of PPBs can be placed. Ohio Trust would like to place this limited number of PPBs in counties so that the allowable branches can reach the maximum possible population. The file Ohio Trust Pop contains the county adjacency matrix described in Problem 15 as well as the population of each county.

- Assume that only a fixed number of PPBs, denoted k. can be established. Formulate a linear binary integer program that will tell Ohio Trust Inc. where to locate the fixed number of PPBs in order to maximize the population reached.
- Suppose that two PPBs can be established. Where should they be located to maximize the population served?
- Solve your model from part a for allowable number of PPBs ranging from 1 to 10. In other words, solve the model 10 times, k set to 1,2, . . . , 10. Record the population reached for each value of k. Graph the results of this analysis by plotting the population reached versus number of PPBs allowed. Based on their cost calculations, Ohio Trust considers an additional PPB to be fiscally prudent only if it increases the population reached by at least 500,000 people. Based on this graph, what is the number of PPBs you recommend to be implemented?

*Hint: *

*Introduce variable yi = 1 if it is possible to establish a branch in county i, and *

*yi =0 otherwise; that is, if county i is covered by a PPB, then the population can be counted as covered.*

Question 4 – 10 Marks

The employee credit union at State university is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue- producing investments together with annual rates of return are as follows:

Types of Load/Investment | Annual Rate of Return (%) |

Automobile Loans | 8 |

Furniture loans | 10 |

Other secured loans | 11 |

Signature loans | 12 |

Risk-free securities | 9 |

The credit union will have $2 million available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments:

- Risk-free securities may not exceed 30% of the total funds available for investment.
- Signature loans may not exceed 10% of the funds invested in all loans (automobile, furniture, other secured, and signature loans).
- Furniture loans plus other secured loans may not exceed the automobile loans.
- Other secured loans plus signature loans may not exceed the funds invested in risk-free securities

- How should the $2 million be allocated to each of the loan/investment alternatives to maximize total annual return?
- What is the projected total annual return?